Integrand size = 28, antiderivative size = 280 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=-\frac {13267820528 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{11402015625}-\frac {400516993 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac {569519 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{28153125}+\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1764163292393 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20730937500 \sqrt {33}}-\frac {13267820528 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5182734375 \sqrt {33}} \]
62/2925*(1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2)+2/75*(1-2*x)^(5/2)*(2+3* x)^(5/2)*(3+5*x)^(5/2)-1764163292393/684120937500*EllipticE(1/7*21^(1/2)*( 1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-13267820528/171030234375*EllipticF( 1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+142391/7239375*(2+3*x )^(3/2)*(3+5*x)^(5/2)*(1-2*x)^(1/2)+3698/482625*(2+3*x)^(5/2)*(3+5*x)^(5/2 )*(1-2*x)^(1/2)-400516993/2533781250*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^( 1/2)-569519/28153125*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-13267820528 /11402015625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 9.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.42 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (12155574323+173484591165 x+48836706750 x^2-528977216250 x^3-336683182500 x^4+621672975000 x^5+547296750000 x^6\right )+1764163292393 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1817234574505 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{684120937500} \]
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(12155574323 + 173484591165* x + 48836706750*x^2 - 528977216250*x^3 - 336683182500*x^4 + 621672975000*x ^5 + 547296750000*x^6) + (1764163292393*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sq rt[9 + 15*x]], -2/33] - (1817234574505*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqr t[9 + 15*x]], -2/33])/684120937500
Time = 0.35 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {112, 27, 171, 27, 171, 27, 171, 27, 171, 25, 171, 27, 171, 25, 176, 123, 129}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {2}{75} \int -\frac {5}{2} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2} (31 x+23)dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \int (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2} (31 x+23)dx+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (\frac {2}{195} \int \frac {1}{2} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2} (1849 x+2656)dx+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \int \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2} (1849 x+2656)dx+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {2}{165} \int \frac {(284773-427173 x) (3 x+2)^{3/2} (5 x+3)^{3/2}}{2 \sqrt {1-2 x}}dx+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \int \frac {(284773-427173 x) (3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}dx+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {1}{45} \int -\frac {3 \sqrt {3 x+2} (5 x+3)^{3/2} (10251342 x+7830965)}{2 \sqrt {1-2 x}}dx\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (10251342 x+7830965)}{\sqrt {1-2 x}}dx+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (1201550979 x+789074087)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \int \frac {(5 x+3)^{3/2} (1201550979 x+789074087)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (53071282112 x+34486181421)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \int \frac {\sqrt {5 x+3} (53071282112 x+34486181421)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (-\frac {1}{9} \int -\frac {1764163292393 x+1116876385759}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \int \frac {1764163292393 x+1116876385759}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {291892051616}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1764163292393}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {291892051616}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1764163292393}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (-\frac {53071282112}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1764163292393}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
(2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/75 + ((62*(1 - 2*x)^(3 /2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/195 + ((3698*Sqrt[1 - 2*x]*(2 + 3*x)^ (5/2)*(3 + 5*x)^(5/2))/165 + ((142391*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5 *x)^(5/2))/15 + ((-10251342*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/3 5 + ((-400516993*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-53071 282112*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-1764163292393*Sqr t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (5307128211 2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9)/10)/ 35)/30)/165)/195)/15
3.28.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 1.37 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-492567075000000 x^{9}-937140435000000 x^{8}-11007171000000 x^{7}+1713392598819 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1764163292393 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+937455630300000 x^{6}+362238910312500 x^{5}-361521647968500 x^{4}-215604575302050 x^{3}+36835025076780 x^{2}+33779897017530 x +2188003378140\right )}{684120937500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(170\) |
| risch | \(-\frac {\left (547296750000 x^{6}+621672975000 x^{5}-336683182500 x^{4}-528977216250 x^{3}+48836706750 x^{2}+173484591165 x +12155574323\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{22804031250 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {1116876385759 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2508443437500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1764163292393 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2508443437500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(272\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {3855213137 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{506756250}+\frac {12155574323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{22804031250}+\frac {1116876385759 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2394423281250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1764163292393 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2394423281250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1669631 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{779625}-\frac {52782 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{3575}-\frac {2239057 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96525}+24 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{6}+\frac {1772 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{65}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(328\) |
-1/684120937500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-49256707500000 0*x^9-937140435000000*x^8-11007171000000*x^7+1713392598819*5^(1/2)*(2+3*x) ^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((10+15*x)^(1/2),1/35 *70^(1/2))-1764163292393*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5 *x)^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+937455630300000*x^6+362 238910312500*x^5-361521647968500*x^4-215604575302050*x^3+36835025076780*x^ 2+33779897017530*x+2188003378140)/(30*x^3+23*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.28 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {1}{22804031250} \, {\left (547296750000 \, x^{6} + 621672975000 \, x^{5} - 336683182500 \, x^{4} - 528977216250 \, x^{3} + 48836706750 \, x^{2} + 173484591165 \, x + 12155574323\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {59943118993271}{61570884375000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1764163292393}{684120937500} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/22804031250*(547296750000*x^6 + 621672975000*x^5 - 336683182500*x^4 - 52 8977216250*x^3 + 48836706750*x^2 + 173484591165*x + 12155574323)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 59943118993271/61570884375000*sqrt(-30 )*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1764163292393/68 4120937500*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPIn verse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\text {Timed out} \]
\[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]